Weak convergence of measure-valued processes and $r$-point functions

TL;DR

This paper establishes conditions under which measure-valued processes converge to super-Brownian motion, focusing on Fourier transforms of $r$-point functions and survival probabilities, with applications to statistical mechanical models.

Contribution

It provides a set of verifiable conditions for convergence of measure-valued processes to super-Brownian motion, linking Fourier transforms and survival probabilities.

Findings

Conditions verified for critical oriented percolation

Conditions verified for critical contact process

Conditions verified for lattice trees at criticality

Abstract

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the $r$-point functions and perhaps convergence of the ``survival probabilities.'' These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.